charging stations placement in drone path planning for
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Charging Stations placement in Drone Path planning forlarge space surveillance
Jean Louis Fendji Kedieng Ebongue, Israel Bayaola, Christopher Thron, AnnaFörster
To cite this version:Jean Louis Fendji Kedieng Ebongue, Israel Bayaola, Christopher Thron, Anna Förster. ChargingStations placement in Drone Path planning for large space surveillance. CARI 2020 - Colloque Africainsur la Recherche en Informatique et en Mathématiques Apliquées, Oct 2020, Thies, Sénégal. �hal-02925708�
Charging Stations placement in Drone Path
planning for large space surveillance
Jean Louis Fendji Kedieng Ebongue*, Israel Bayaola*, Christopher
Thron**, and Anna Förster***
* University of Ngaoundéré – CAMEROON, [email protected], [email protected]
** Texas A&M University Central Texas – USA, [email protected]
*** University of Bremen – Germany, [email protected]
RÉSUMÉ. Les stations de charge ont récemment été introduites pour assister les drones dans les
missions de surveillance de longue durée, par exemple dans les grandes exploitations agricoles. Mais
le coût de ces équipements supplémentaires reste un obstacle important à leur adoption. Il est donc
impératif d'en minimiser leur nombre lors de la planification de la trajectoire du drone. A cette fin, ce
travail formule le problème du drone unique avec plusieurs stations de charge (SD-MCS). Dans cette
formulation, une zone d'intérêt à couvrir est donnée ainsi qu'un ensemble d'emplacements potentiels
de stations de charge où le drone peut atterrir pour recharger sa batterie. L'objectif est de minimiser
le nombre de stations de charge effectivement utilisées et ensuite le temps d'exécution de la mission.
Trois approches sont proposées pour résoudre le problème. Un ensemble de 30 topologies aléatoires
a été généré pour évaluer les différents algorithmes. Après simulations, le Back and Forth Simulated
Annealing donne des résultats satisfaisants avec un nombre optimal de stations de charge.
ABSTRACT. Charging stations have recently been introduced to assist drones in long missions’
surveillance, such as in large farms. However, the cost of such extra equipment remains a significant
barrier to their adoption. It is therefore imperative to minimize the number of charging stations during
the path planning of the drone. To this end, this work formulates the Single Drone with Multiple
Charging Stations problem (SD-MCS). In this formulation, an area of interest to be covered is given
as well as a set of potential charging station locations where the drone can land to recharge its battery.
The aim is primarily to minimize the number of locations for charging stations, and secondarily to
minimize the completion time of the surveillance mission. Three approaches are proposed to solve
the problem. A set of 30 random topologies has been generated to test the different algorithms. From
computational experiments, Back and Forth Simulated Annealing provides satisfactory results with
an optimal number of charging stations.
MOTS-CLÉS : Planification de la trajectoire du drone, stations de charge, aller-retour, recuit simulé.
KEYWORDS: Drone Path Planning, Charging Stations, Back and Forth, Simulated Annealing.
Proceedings of CARI 2020 Bruce Watson, Eric Badouel, Oumar Niang, Eds.
Ecole Polytechnique de Thiès, Sénégal October 2020
1. Introduction
Although at one time drones were used only by the military sector, they are now used in
several civilian applications areas such as smart farming[1], disaster management[2], and
surveillance[3]. Earlier drones were manually and remotely operated by humans, but now
flights can be pre-programmed and the drone’s mission executed automatically. In this
last mode, the path of the drone should be determined before the mission: the task of prior
path determination is known as Coverage Path Planning[4].
Coverage Path Planning can be defined as follows: Given an area of interest, determine
an optimal path of the drone allowing it to complete its mission of covering the area of
interest. The objective is usually to reduce the completion time of the mission. The area
of interest can be decomposed or not, depending on its regularity or its complexity. In
most cases, it is decomposed into cells based on the Ground Sampling Distance (GSD),
which is defined as the distance between pixel centers measured on the ground [5].
An important issue when planning the path of a drone is the energy limitation due to the
small capacity of the battery. Huge amount of works tackling CPP proposed energy-aware
algorithms based on some observations, mainly the fact that a drone spends a lot of time
and energy making turns. Therefore, those algorithms modify conventional trajectories
such as Back-and-Forth [6] and HILBERT[7]. However, the limited capacity of drones
prevents them to perform large space missions. An interesting survey on coverage path
planning with drone can be found in [8].
To assist drones for large-area missions, charging stations have recently been introduced
[9]. When a drone runs out of energy, it lands on a charging station to recharge its battery.
Scenarios in [10] use pre-defined charging station locations with only one drone. The aim
is to minimize the mission completion time by deciding when the drone should stop for
charging depending on the state of charge (SoC). Authors in [11] rather consider a fleet
of drones and tackle the problem of coverage and energy replenishment scheduling with
only one charging station at the center of the scenario. This limits the area to a radius of
half the traveling distance of a drone (round trip).
To automatically surveil larger areas will require a greater number of drones. How to
minimize their number and determine their optimal locations? In fact, a greater number
of charging stations drastically increase the cost of the architecture for automatic
surveillance. For instance, the charging pad from Heisha costs around USD 1000 while a
Mavic Air drone that can be charged using this pad costs around USD 600 [12]. It is
Proceedings of CARI 2020
therefore important to minimize the number of charging station in order to reduce the cost
of the architecture, especially in presence of limited budget as it is the case for low-
income users.
This work introduces the Single Drone and Multiple Charging Station (SD-MCS)
problem. Unlike previous work focusing on the minimization of the completion time of
the mission, this work rather tackles drone path planning with a priority on the economical
point of view. In this formulation, an area of interest to cover is given as well as a set of
potential locations for charging stations where the drone can land to recharge its battery.
The aim is first to minimize the number of effectively used charging stations and then the
completion time of the surveillance mission. To reach this goal, three approaches have
been defined: the well-known Back and Forth (BF), Simulated Annealing (SA), and the
combination of the two previous approaches namely Back and Forth Simulated Annealing
(BFSA). The latter is based on the assumption that the quality of the initial solution of
SA can influence the final solution. Thus, SA is combined with BF. BF generates the
initial solution that is later improved by SA. To test the proposed approach, a set of 30
random topologies has been generated.
The rest of the paper is organized as follows: Section 2 presents the system model and
the formulation of the Single Drone and Multiple Charging Station problem. Section 3
presents the different optimization approaches defined to solve the problem. Section 4
presents the simulation setup and compares the results. This paper ends with a conclusion
and future work.
2. System Model
2.1. Scenario modelling
The area to cover is considered as a rectangular area of size 𝐷 m². It is decomposed
into cells as shown in Figure 1. Each cell has a unique integer identifier. The set of the
cells of area is represented by 𝐴 and contains 𝑛 × 𝑚 elements. Some cells are candidate
locations for charging station deployment where drone can stop to recharge its battery.
Their set is represented by 𝐶, with 𝐶 ⊆ 𝐴. There is only one drone with a maximal
capacity energy 𝐸𝑀. When the drone stops at charging station, it recharges its battery to
𝐸𝑀 before leaving. The drone starts its mission at the take-off point 𝑆𝑡 and ends it at the
landing point.
The following assumptions have been made: the drone starts with the maximum
energy, but this energy cannot allow it to cover the whole area; the area to be covered has
Charging Stations Placement in Drone Path Planning for Large Space Surveillance
no obstacles; the wind speed is constant over the whole area; the drone flies at a constant
height; and finally the ambient temperature, the taking off energy and landing energy are
neglected.
2.2. Energy model
The power consumed during horizontal flight and cover has been considered
approximately equivalent [13]. But it is not the case in the reality. For a better
approximation, we use the energy model proposed by [10]. This model is based on
empirical experiments with an estimation error around 0.4%. The power and energy are
estimated respectively by (1) and (2).
�̂� = [
𝛽1
𝛽2
𝛽3
]
𝑇
[
‖𝑣𝑥𝑦⃗⃗ ⃗⃗ ⃗⃗ ‖
‖𝑎𝑥𝑦⃗⃗ ⃗⃗ ⃗⃗ ‖
‖𝑣𝑥𝑦⃗⃗ ⃗⃗ ⃗⃗ ‖‖𝑎𝑥𝑦⃗⃗ ⃗⃗ ⃗⃗ ‖
] + [
𝛽4
𝛽5
𝛽6
]
𝑇
[
‖𝑣𝑧⃗⃗ ⃗‖
‖𝑎𝑧⃗⃗⃗⃗ ‖
‖𝑣𝑧⃗⃗ ⃗‖‖𝑎𝑧⃗⃗⃗⃗ ‖
] + [
𝛽7
𝛽8
𝛽9
]
𝑇
[
𝑚𝑣𝑥𝑦⃗⃗ ⃗⃗ ⃗⃗ . 𝑤𝑥𝑦⃗⃗ ⃗⃗ ⃗⃗ ⃗
1] (1)
𝐸 = �̂�𝐷 (2)
Where:
Figure 1: Problem description (a) and area decomposition (b)
Proceedings of CARI 2020
▪ 𝑣𝑥𝑦⃗⃗ ⃗⃗ ⃗⃗ and 𝑎𝑥𝑦⃗⃗ ⃗⃗ ⃗⃗ indicate the speed and acceleration vectors describing the horizontal
movement of the drone in m/s;
▪ 𝑣𝑧⃗⃗ ⃗ and 𝑎𝑧⃗⃗⃗⃗ the speed and acceleration vectors describing the vertical movement of the
drone in m/s2;
▪ 𝑚 the weight of payload in Kg;
▪ 𝑤𝑥𝑦⃗⃗ ⃗⃗ ⃗⃗ ⃗ the vector of wind movement in the horizontal surface;
▪ 𝛽1, … , 𝛽9 are coefficients determined empirically;
▪ 𝐷 is the duration.
2.3. Single Drone and Multiple Charging Station problem
formulation
Formally, the SD-MCS is defined as follows: given an area to cover decomposed into
a set of cells 𝐴 and a set of potential locations for charging station 𝐶, determine a route 𝑅
that will minimize the number of effectively used charging stations and the energy
consumption of the mission. A route 𝑅 is nothing else than a permutation of the elements
of 𝐴. More specifically, 𝑅 = (𝑅1, 𝑅2, … , 𝑅𝑛×𝑚) with 𝑅𝑖 ∈ 𝐴, 1 ≤ 𝑖 ≤ 𝑛 × 𝑚.
The objective functions are given by (3) and (5). They respectively minimize the
number of charging station and the energy consumption.
𝑀𝑖𝑛 𝑓1(𝑅) = ∑ℎ(𝑅𝑖)
𝑛
𝑖=1
(3)
𝑀𝑖𝑛 𝑓2(𝑅) = ∑𝑃(𝑅𝑖, 𝑅𝑖+1)
𝑛
𝑖=1
𝑡𝑅𝑖,𝑅𝑖+1 (4)
Where:
ℎ(𝑅𝑖) = {1, 𝑅𝑖 ∈ 𝐶0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
(5)
▪ 𝑛 = 𝐶𝑎𝑟𝑑(𝑅);
▪ 𝑅𝑖 ∈ 𝑅;
▪ 𝑃(𝑅𝑖 , 𝑅𝑖+1) is the power consumed by the drone between 𝑅𝑖 and 𝑅𝑖+1 given by eq.
(1);
▪ 𝑡𝑅𝑖,𝑅𝑖+1 is the time took by the drone to go from point 𝑅𝑖 to point 𝑅𝑖+1.
Charging Stations Placement in Drone Path Planning for Large Space Surveillance
3. Optimization approaches
3.1. Back and Forth
Back and Forth is the basic technique used especially in regular surfaces. It is inspired
by what is called in the literature "the way of the ox", the approach is the following: when
an ox trailed a plow and must cover an area, it travels along the field in a straight line,
turns around, then makes a new rectilinear path adjacent to the previous one. The
approach is given by Algorithm1.
Algorithm 1: Back and Forth (BF)
Input: 𝑓1, 𝑓2: Objective functions to minimize
Output: 𝑅: A route
1
2
3
4
5
6
7
8
9
10
11
12
Begin
k:=0, R:=[ ]
for j from 0 to n:
for i from 0 to m:
if j mod 2 = 0:
p ≔i × n + j + 1
else:
p ≔ (m-i-1) ×n + j + 1
R[k] ≔p
k ≔k + 1
return R
End
3.2. Simulated Annealing
The SA algorithm is inspired from the annealing process in metallurgy. It proceeds as
follows: an initial solution is generated, then several iterations are performed to improve
this solution. A neighbour solution is generated from the previous one at each iteration.
The new solution is accepted if it improves the value of the objective function. If it is not
the case, the new solution is accepted with a probability depending on the current
temperature 𝑇 and the difference between the value of its objective function and the one
of the previous solution ∆𝐸. The probability in the SA algorithm generally follows the
Boltzmann distribution given by 𝑒−∆𝐸
𝑇 . Since the temperature updates each time the
equilibrium state is reached, the probability of accepting non-improving solutions
decreases. The algorithm stops when a minimal temperature is reached. A particularised
version of SA is given in Algorithm 2.
Proceedings of CARI 2020
Algorithm 2: Simulated Annealing (SA)
Input: 𝑓1 : first objective function (number of charging station)
𝑓2 : second objective function (energy consumption)
Output: 𝑅: the best route found
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Begin
𝑇 ∶= 𝑇𝑖𝑛𝑖𝑡𝑖𝑎𝑙
𝑅:= initialSolution()
𝑣1:= 𝑓1(𝑅); 𝑣2:= 𝑓2(𝑅)
while (stopping condition not met) do
while (equilibrium condition not met) do
𝑅’ := newSolution()
𝑣1’ := 𝑓1(𝑅’); 𝑣2’ := 𝑓2(𝑅’);
∆𝐸1 ∶= 𝑣’1 − 𝑣1
∆𝐸2 ∶= 𝑣’2 − 𝑣2
if ∆𝐸1 < 0 or ( ∆𝐸1 = 0 and ∆𝐸2 < 0 ) then 𝑅 ∶= 𝑅’
else accept R’ if ∆𝐸1 = 0 with probability 𝑒−∆𝐸2
𝑇
Update(𝑇)
return 𝑅
End
InitialSolution (line 3): The initial route is obtained by generating a permutation of
the cells, starting by the defined take-off point and ending by the defined landing point.
Each cell has between 3 to 8 Moore neighbours. Each time a cell is included in the route,
the next cell is chosen in the Moore neighbourhood of the previous cell. If it is not
possible, then a random cell is generated.
Fitness functions (lines 4): the two objective functions namely 𝑓1 and 𝑓2 are evaluated
using respectively eq. (3) and (4).
NewSolution (line 7): It is obtained from the current solution using a k-opt operator.
A k-opt operator permutes k edges. A 4-opt operator is used with greater temperatures
while a 2-opt is used when the temperature is close to the threshold. An illustration of the
2-opt operator application on a given route is given in Figure 2.
Stopping condition (line 5): If the current temperature is lower than a certain
threshold, or the we suppose therefore having reached the optimal.
Equilibrium stage (line 6): If there is no improvement of the best solution found so
far after a certain number of iteration, we suppose having reached an equilibrium state
and we update the temperature.
Charging Stations Placement in Drone Path Planning for Large Space Surveillance
Accepting a solution (line 11 and 12): A solution is accepted if it provides a smaller
number of charging station, or if the number of charging station is the equal to the one of
the best solution found so far, but with a smaller energy consumption. However, a non-
improving solution with regard to energy consumption can be accepted in order to escape
from local optima, but with a certain probability.
Figure 2: Application of 2-opt operator. (a) initial route: (b) neighbor route by permuting
edges (2,3) and (5,6).
3.3. Back and Forth Simulated Annealing (BFSA)
BFSA is a combination of BF and SA. It is based on the idea that the initial solution
of SA can improve its result. So, the initial solution is generated by BF and it is improved
using SA.
4. Performance evaluation
4.1. Parameters setting
We use the parameters of the drone 3DR Solo. It has a maximum energy of 76.96𝑊ℎ.
We considered that the drone flies at a speed of 0.5𝑚/𝑠 and at an altitude of about 120𝑚.
𝛽 parameters used the power in eq. (1) for 3DR solo drone are the following [10]: 𝛽1 =−1.526, 𝛽2 = 3.934, 𝛽3 = 0.968, 𝛽4 = 18.125, 𝛽5 = 96.613, 𝛽6 = −1.085, 𝛽7 =0.220, 𝛽8 = 1.332, 𝛽9 = 433.9. The wind speed is set to 3.88m/s.
We generated a set of 30 random topologies of size 4 × 4 𝑐𝑒𝑙𝑙𝑠 (i.e 𝑛 = 𝑚 = 4), with
4 potential charging locations. We performed 30 executions of each approach on each of
these instances. Approaches have been compared with regards to three metrics namely:
the feasibility, the number of charging station, and the energy consumption. A solution is
a feasible if the maximum energy of the drone allows it to fly from the starting point or
an intermediary charging station to the closest charging station on the route, or from the
last charging station to the final landing point. All the algorithms have been implemented
in Python.
Proceedings of CARI 2020
4.2. Simulation results and discussion
Table 1 summarizes the results of the simulation. The data are the mean values of all
the 30 runs for each instance. The best results are in bold. From the results, BFSA
provides the best feasibility with a total average of 99.55% compared to 87.22% for SA
and only 33.33% for BF. More specifically, BFSA provides a 100% feasibility on 27 out
30 instances. The distribution of potential location for charging stations does not allow
BF solution to be feasible on 20 out 30 instances.
Regarding the number of charging station, BFSA again provides the best result with
a total average of 3.02 charging stations, compared to 3.1 for BF and 3.14 for SA.
Explicitly, BFSA provides the minimum number of charging station for all instances
excerpt instance 2 where BFSA provides 3.43 compared to 3.37 for SA. But when
observing the feasibility metric for that instance, we realize that SA provides 66.67%
compared to 93.33% for BFSA. That means, SA provides only 10 feasible solutions out
of 30 runs when BFSA provides 28. This shows that BFSA find a feasible solution even
at the expense of the number of charging station, where SA ends without finding a feasible
solution. The average in terms of number of charging station of BFSA can therefore be
greater than the one SA for instance 2.
Figure 3: Mean Energy consumption
The last metric is the energy consumption. Wherever BF finds a feasible solution, the
energy consumption is the smallest. But as discussed earlier, BF found only 10 feasible
solutions out of 30 instances. Wherever BF finds a feasible solution, BFSA provides the
same energy consumption except for instance 28 where BFSA provides 251,92Wh
Charging Stations Placement in Drone Path Planning for Large Space Surveillance
compared to 234,05Wh for BF. But a look to the number of charging station shows that
BFSA requires only 3 charging stations compared to 4 for BF. Since the priority is to
reduce the number of charging station, BFSA’ solution is considered as the best. In fact,
if we considered the maximum energy of a 3DR Solo drone that is 76.96Wh, we will need
at least 4 recharges to provides the required 234,05Wh. Since the drone starts with a full
charge, only 3 intermediary charging stations are necessary.
To better observe the energy consumption of the different approaches, we plotted their
mean values in Figure 3. From Figure 3, BFSA provides a smaller energy consumption
compared to SA.
With regards on all the three metrics the difference between BFSA and SA
demonstrate the impact of the initial solution on SA.
Table 1: Simulation results summarization (mean values)
Feasibility Number of Charging Station Energy
BF SA BFSA BF SA BFSA BF SA BFSA
Inst1 0 96,67 100 _ 3,03 3 _ 265,48 253,40
Inst2 0 66,67 93,33 _ 3,37 3,43 _ 300,76 288,87
Inst3 0 80 96,67 _ 3,27 3,07 _ 292,15 273,55
Inst4 0 100 100 _ 3 3 _ 264,37 256,76
Inst5 100 96,67 100 3 3,07 3 234,05 274,56 234,05
Inst6 0 80 100 _ 3,20 3 _ 282,34 253,51
Inst7 0 100 100 _ 3 3 _ 278,41 256,88
Inst8 0 86,67 100 _ 3,13 3,03 _ 273,89 259,54
Inst9 100 86,67 100 3 3,17 3 234,05 283,19 234,05
Inst10 0 93,33 100 _ 3,13 3,03 _ 279,21 284,08
Inst11 100 93,33 100 3 3,07 3 234,05 261,92 234,05
Inst12 100 93,33 100 3 3,07 3 234,05 267,67 234,05
Inst13 0 93,33 100 _ 3,07 3 _ 284,84 264,13
Inst14 0 83,33 100 _ 3,20 3 _ 287,25 261,10
Inst15 0 86,67 96,67 _ 3,13 3,03 _ 287,38 277,86
Inst16 100 90 100 3 3,10 3 234,05 276,76 234,05
Inst17 0 63,33 100 _ 3,40 3,03 _ 303,81 278,47
Inst18 100 100 100 3 3 3 234,05 267,99 234,05
Inst19 0 63,33 100 _ 3,37 3 _ 298,77 260,16
Inst20 0 80 100 _ 3,20 3 _ 285,00 256,18
Proceedings of CARI 2020
Inst21 0 100 100 _ 3 3 _ 275,00 253,92
Inst22 0 90 100 _ 3,10 3 _ 281,34 259,92
Inst23 100 93,33 100 3 3,07 3 234,05 271,52 234,05
Inst24 0 73,33 100 _ 3,27 3 _ 292,62 256,79
Inst25 100 83,33 100 3 3,17 3,03 234,05 283,19 234,05
Inst26 0 93,33 100 _ 3,10 3 _ 282,67 262,11
Inst27 100 83,33 100 3 3,20 3,03 234,05 276,29 234,05
Inst28 100 93,33 100 4 3,13 3 234,05 278,45 251,92
Inst29 0 96,67 100 _ 3,03 3 _ 264,37 260,40
Inst30 0 76,67 100 _ 3,23 3 _ 285,82 257,02
Total Average 33,33 87,22 99,55 3,1 3,14 3,02 234,05 280,23 254,43
6. Conclusion and future work
This paper has introduced the Single Drone with Multiple Charging Stations problem
in which the objectives are the minimization of the number of charging station and the
energy consumption of the drone. Three approaches have been proposed namely BF, SA
and BFSA. Simulation results have shown the superiority of BFSA over the two others.
In fact, BFSA almost always finds a feasible solution even when the others could not. In
addition, BFSA provides the least expensive architecture due to the minimum number of
charging station close to the optimal solution. Finally, BFSA provides in general the best
energy-efficient solution.
Further investigation will be conducted to consider more realistic parameters such as
wind speed and direction changing, obstacles and No-Fly Zone (NFZ). Moreover, a web
interface will be designed to allow user to interact with the system directly on maps and
to generate waypoints that can be imported into the drone.
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